Is there a way to restrict Brownian Motion to a specific axis?
This solution assumes that the motion is introduced by the Force Fields on planes. The emitter only has some Normal value to kick start the motion. Gravity is turned off.



You could use a Plane object set to a Harmonic Field (shape: Surface) with strength 4, rest length 0.2. I placed mine above the emitter plane.



To get a random pattern you could vary the emission of particles by using a texture. To get randomness in the oscillation you could use 2 opposing planes with Lennard Jones effect type. Using large values, they will juggle particles between themselves. Any that are spilled come to rest.



The plane at the bottom is the emitter.